iSoul Time has three dimensions

Coordinate transformations

Coordinate Transformations with t

r = space coordinate, t = time coordinate, v = velocity, u = pace

 

Galileian transformation

speed: = rvt and t′ = t,         pace: = rt/u and t′ = t.

Co-Galileian transformation

speed: = tr/v and r′ = r,       pace: = tur and r′ = r.

 

Light:   c := speed of light,        ç := pace of light.

speed: r = ct or r/c = t and r′ = ct′, or r′/c = t′,

pace:  çr = t or r = t/ç and çr′ = t′ or r′ = t′/ç.

 

Lorentz transformation

speed: γ = (1 – v2/c2)–1/2 with r′ = γ (r − vt) and t′ = γ (trv/c²),

pace:  γ = (1 – ç2/u2)–1/2 with r′γ (rt/u) and t′γ (trç²/u),

which applies only if |v| < |c| or |u| > |ç|.

 

Co-Lorentz transformation

speed: λ = (1 − c2/v2)–1/2 with t′λ (t − r/v) and r′λ (rt (c2/v)),

pace:  λ = (1 – u2/ç2)–1/2 with t′λ (tur) and r′ λ (rt (u/ç²)),

which applies only if |v| > |c| or |u| < |ç|.

Galilei-Lorentz limit

speed: c1 → ∞: γ1 → 1, t′ → t,                c2c/2: γ2 → (1 – 4v²/c²)−1/2 and t′ → γ2 (t – 4r (v/c²)).

pace:  ç1 → 0: γ1 → 1, t′ → t,                 ç2 → 2/ç: γ2 → (1 – (1/4) ç²/u²)−1/2 and t′ → γ2 (t – (r/4) (ç²/u)).

Co-Galilei-Lorentz limit

speed: c1 → 0: λ1 → 1, r′ → r,               c2 → 2/c: λ2 → (1 – (1/4) c²/v²)−1/2 and r′ → λ2 (r – (t/4) (c²/v)).

pace:  ç1 → ∞: λ1 → 1, r′ → r,               ç2ç/2: λ2 → (1 – 4u²/ç²)−1/2 and r′ → λ2 (r – 4t (u/ç²)).

 

If |v| = |c|, then r′ = r and t′ = t.


Coordinate Transformations with τ

r = space coordinate, t = time coordinate, v = velocity, u = pace

c := speed of light,        ç := pace of light.

τ := ct                           τ := t/ç

 

Galileian transformation

speed: = rτ (v/c) and τ′ = τ,              pace: = rτ (ç/u) and τ′ = τ.

Co-Galileian transformation

speed: τ´ = τr (c/v) and r′ = r,              pace: τ´ = τr (u/ç) and r′ = r.

 

Light: r = τ and r′ = τ′

 

Lorentz transformation

speed: γ = (1 – v2/c2)–1/2 with r′ = γ (r − τ (v/c)) and τ′ = γ (τr (v/c)),

pace:  γ = (1 – ç2/u2)–1/2 with r′γ (rτ (ç/u)) and τ′γ (τr (ç/u)),

which applies only if |v| < |c| or |u| > |ç|.

 

Co-Lorentz transformation

speed: λ = (1 − c2/v2)–1/2 with τ′λ (τ – r (c/v)) and r′λ (rτ (c/v)),

pace:  λ = (1 – u2/ç2)–1/2 with τ′λ (τr (u/ç)) and r′ λ (rτ (u/ç)),

which applies only if |v| > |c| or |u| < |ç|.

 

Galilei-Lorentz limit

speed: c1 → ∞: γ1 → 1, τ′ → τ,               c2c/2: γ2 → (1 – 4v²/c²)−1/2 and τ′ → γ2 (τ – 4r (v/c)).

pace:  ç1 → 0: γ1 → 1, τ′ → τ,                 ç2 → 2/ç: γ2 → (1 – (1/4) ç²/u²)−1/2 and τ′ → γ2 (τ – (r/4) (ç/u)).

Co-Galilei-Lorentz limit

speed: c1 → 0: λ1 → 1, r′ → r,               c2 → 2/c: λ2 → (1 – (1/4) c²/v²)−1/2 and r′ → λ2 (r – (τ/4) (c/v)).

pace:  ç1 → ∞: λ1 → 1, r′ → r,               ç2ç/2: λ2 → (1 – 4u²/ç²)−1/2 and r′ → λ2 (r – 4τ (u/ç)).

 

If |v| = |c|, then r′ = r and τ′ = τ.


Coordinate Transformations with c := 1

r = space coordinate, t = time coordinate, v = velocity, u = pace

 

Galileian transformation

speed: = rvt and t′ = t,         pace: = rt/u and t′ = t.

Co-Galileian transformation

speed: = tr/v and r′ = r,       pace: = tur and r′ = r.

 

Light:   1 = c := speed of light,  1 = ç := pace of light.

speed: r = t and r′ = t′

pace:  r = t and r′ = t′.

 

Lorentz transformation

speed: γ = (1 – v2)–1/2 with r′ = γ (r − vt) and t′ = γ (trv),

pace:  γ = (1 – 1/u2)–1/2 with r′γ (rt/u) and t′γ (tr/u),

which applies only if |v| < 1 or |u| > 1.

 

Co-Lorentz transformation

speed: λ = (1 − 1/v2)–1/2 with t′λ (t − r/v) and r′λ (rt (1/v)),

pace:  λ = (1 – u2)–1/2 with t′λ (tur) and r′ λ (rtu),

which applies only if |v| > 1 or |u| < 1.

 

If |v| = 1, then r′ = r and t′ = t.

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